1
vote
0answers
16 views

A question on Lagrange multipliers

The state of Megalomania occupies the region $x^4 + y^4 \leq 30,000.$ The altitude at the point $(x,y)$ is $\frac{1}{8}xy+200x$ meters above sea level. Where are the highest and lowest points in the ...
0
votes
1answer
28 views

Optimization with a constraint given by a differential equation

I have the following differential equation $$\ddot\theta(t) = -k\sin{\omega t}\sin{\theta(t)} \quad \text{where} \quad \theta(0)=\theta_0, \dot\theta(0)=v_0$$ where $\omega$ is a known constant and ...
-2
votes
1answer
34 views
5
votes
2answers
67 views

When is $\min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))$?

When is $$ \min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))? $$
1
vote
1answer
37 views

Constrained Optimization of a function of two variables.

I was given the following tutorial problem, and I'm having a bit of trouble seeing how it works. I've been asked to find the four critical points of this system, with two of these being degenerate ...
0
votes
0answers
23 views

When does a polynomial have finitely many critical points on a level set of another polynomial?

Suppose I have two polynomial functions $f$ and $g$ and I am interested in the critical points that $f$ has on a level set of $g$, i.e. $\{x\in \mathbb R^n : g(x)=a_1\}$ for some $a_1\in \mathbb R$ . ...
1
vote
1answer
35 views

Finding the critical points in a constrained optimization problem using the Lagrangian

I've been given the following constrained optimization problem, but I'm having trouble even getting the critical points out - the numbers just seem way too complicated... Find the local maxima and ...
2
votes
0answers
35 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
0
votes
1answer
48 views

Minimum of function of x and y

any ideas how to find minimum of the following function: $f(x,y)=a-b\frac{x}{y}-c\frac{y}{x}+\frac{d}{x}+\frac{e}{y}-\frac{g}{xy}$. Assume that $a,b,c,d,e,g>0$. We can also assume that $x,y \ge 1$. ...
2
votes
2answers
68 views

System of equations in Lagrange multiplier problem

Continuing from Confounding Lagrange multiplier problem: I'm having trouble solving the system of equations below arisen from a Lagrange multiplier problem where we are to optimize $f(x,y,z) = 4x^2 + ...
1
vote
1answer
41 views

Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
0
votes
0answers
27 views

How many max/min/saddle points of $f(x, y)$ on a region?

Consider the function $f(x,y)=2xe^x\sin y$ on the region $\{(x,y) \mid -\frac{\pi}{4} \leq y \leq \frac{3\pi}{4}\}$. How many maximums/minimums/saddle points are there? I honestly have no idea how ...
0
votes
1answer
20 views

Maximising a function under a constraint

Let $$f(x,y,z) = 4x+2y+5z^2 \text{ and } A=\{(x,y,z) \in \mathbb{R^3} ; \, x^2+y^2+z^4 \leq 5 \}.$$ Find the maximum of $f$ on $A$. My question is the following: How do I prove that the maximum must ...
2
votes
2answers
67 views

Why does this vector derivation hold?

I have the following variables/matrices: $$A \in \mathbb{R}^{m \times n} , \quad p \in \mathbb{R}^{n}, \quad \Sigma \in \mathbb{R}^{m \times m}, \quad w \in \mathbb{R}^{m}$$ where $\Sigma$ is a ...
2
votes
1answer
73 views

How can I find the point (X, Y, Z) which minimizes this quantity?

I have a number of equally powerful light sources $L_i, 1 \le i \le N$ at points within a cube $(x_i, y_i, z_i), -1 \le x_i, y_i, z_i \le 1$. The intensity of each light falls off with distance ...
-1
votes
0answers
32 views

Constraint optimization with lagrangian

I am having trouble regarding the general steps one needs to take in order to solve an constraint optimization using Lagrangian. More specifically, I want to maximize objective equation $f(x,y,z,w)$ ...
1
vote
0answers
56 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
1answer
32 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
1
vote
2answers
38 views

A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
-1
votes
1answer
22 views

Nearest and farthest point from a function to another [closed]

Find the nearest and farthest point from the ellipse $ x^2 + 3y^2 =3 $ to the segment made by $ x+y = 3 $ in the first quadrant. Found in a multivariable calculus course. So I have to find the ...
1
vote
2answers
36 views

Finding extremal values on a set

Let $f(x,y)=(x-1)^2+y^2+xy$. Find the maximal and minimal values of $f$ on the set $M=\{(x,y):|x|+|y|\leq4\}$. Attempt: By taking partial derivatives and solving the homogenous algebraic system we ...
4
votes
1answer
41 views

Local minimum and gradient [duplicate]

But the proof here below is specially elegant. Is there any function $f$ such that $f$ has a local minimum at $x$ but $\nabla f(x) \neq 0$? Only assumption on $f$ is that it has to be differentiable ...
0
votes
2answers
32 views

How to find the absolute extrema of a function on an elliptical cylinder using Lagrange multipliers?

Optimize the function $ f(x,y) = x^2y $ on the elliptical cylinder $ \ x^2 \ + \ 2y^2 \ \le \ 6 \ $ using Lagrange Multipliers. Well, from what I know that I have to find the gradient then to ...
3
votes
2answers
148 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
0
votes
1answer
38 views

A function with positive Hessian at a critical point, without having a minimum there

I have a problem with a little instance: $f(x,y) = \begin{cases} (x^4-3x^2y^2+y^2)/(x^2+y^2) & otherwise \\ 0 & \text{(x,y)=(0,0)} \end{cases}$ This is a example of a function which ...
1
vote
1answer
37 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
1
vote
1answer
20 views

Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
1
vote
0answers
33 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
0
votes
1answer
40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
0
votes
0answers
28 views

Proof that feature normalization cause faster convergence of gradient descent

How to prove that if I do feature normalization (scaling of the $x_1,\ldots,x_n$ to be all in range $[0,1]$) to a convex function $f(x_1,\ldots,x_n)$ that returns real scalar, then gradient descent ...
0
votes
0answers
12 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
0
votes
3answers
42 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
1
vote
1answer
39 views

extrema of funcion

$f(x,y,z)=x+2z$ and $M=\{[x,y,z]\in\mathbb{R}^3:x^2+2y^2=4,z+y\le 1\}$. I found out that M is not bounded from below so it does not have minimum or infimum. But how do I find maximum? I tried to use ...
0
votes
1answer
64 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
1
vote
1answer
38 views

Use Lagrange Multipliers to determine max and min

Using Lagrange Multipliers, determine the maximum and minimum of the function $f(x,y,z) = x + 2y$ subject to the constraints $x + y + z = 1$ and $y^2 + z^2 = 4$: Justify that the points you have found ...
0
votes
1answer
22 views

Gradient descent with adaptive learning ratio.

I have a neural network, trained with SGD (stochastic gradient descent) with learning ratio $\alpha$. Each iteration I try to recalculate the weights with a rule: $$\Delta \vec{w} = -\alpha ...
0
votes
1answer
48 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
1
vote
1answer
50 views

Maximum distance from the origin to the surface

I am having trouble getting the maximum distance from the origin to the surface $$ \frac{x^4}{16} +\frac{y^4}{81} + z^4 = 1 $$ Knowing I have to maximize $x^2 +y^2+ z^2$ and that the constrain ...
1
vote
1answer
28 views

Optimization of a Sum of Variables

Let there be variables $A$, $B$, $C$, $D$, and $E$ such that a total of $N$ points is allocated among the variables: $A$+$B$+$C$+$D$+$E$=$N$, $N$∈$ℝ$. Let the corresponding point values returned by ...
1
vote
0answers
45 views

How to solve Max under an integral?

This is the first time I come accross a Max function inside an integral. I have looked around online and did not find anything about it. I would like to know the rules of what can I do when I have an ...
2
votes
3answers
57 views

Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers

Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers. So I set it up: $$ 1 = 2x\lambda_1 + 2\lambda_2 \\ 1 = -2y\lambda_1 \\ 1 = \lambda_2 $$ Plug ...
1
vote
1answer
32 views

Finding extrema with multiple constraints without Lagrange multipliers

Find the maximums and minimums of $z = 15x+14y$ with constraints $0 \leq x \leq 10, 0 \leq y \leq 5, 3x+2y \geq 6$ I obviously can't take the partial derivatives of inequalities, so I'm at a loss ...
0
votes
1answer
26 views

Image of an unbounded set in $\mathbb R^2$ under the function $f(x,y)=x^3+4y^2-4xy$

Given the function $f(x,y)=x^3+4y^2-4xy$ to be evaluated over the set $E={(x,y) \in R^2: 0\leq y \leq 3x/4}$, I'm asked to determinate $F(E)$. I've noticed that the function in continuous, and the ...
2
votes
3answers
50 views

Find max/min of $e^{2x}\left(x+y^{2}+2y\right)$

$$e^{2x}\left(x+y^{2}+2y\right)$$ FOC: $$\begin{cases} 2e^{2x}\left(x+y^{2}+2y\right)+e^{2x}=0\\ e^{2x}\left(2y+2\right)=0 \end{cases}\rightarrow\begin{cases} x=\frac{1}{2}\\ y=-1 \end{cases}$$ SOC: ...
2
votes
1answer
83 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
1
vote
1answer
33 views

Locally minimizing a concave function

What will happen if we minimize a concave function via gradient descent? Where does it get stuck? Intuitively a concave function has more structure than an arbitrary function, and seem to be easier ...
2
votes
2answers
54 views

Point on $z = \frac{1}{xy}$ closest to origin

Where $x>0$ and $y>0$. I want to work with the square of the distance formula from the origin, so I went with $f(x,y) = x^2 + y^2 + \frac{1}{(xy)^2}$. Then I found the first partial ...
2
votes
1answer
84 views

The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
2
votes
2answers
74 views

Gradient and Swiftest Ascent

I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$) The standard proof is to note that the ...
0
votes
2answers
85 views

Distance from Ellipsoid to Plane - Lagrange Multiplier

Find the distance from the ellipsoid $x^2 + y^2 + 4z^2 = 4$ to the plane $x + y + z = 6$. I'm trying to do it using Lagrange multipliers over the distance equation, but then it just gets ...